Step |
Hyp |
Ref |
Expression |
1 |
|
hstosum |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) = ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) |
2 |
1
|
fveq2d |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ) = ( normℎ ‘ ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ) |
3 |
2
|
oveq1d |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ) ↑ 2 ) = ( ( normℎ ‘ ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ↑ 2 ) ) |
4 |
|
hstcl |
⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ ) |
5 |
4
|
adantr |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( 𝑆 ‘ 𝐴 ) ∈ ℋ ) |
6 |
|
hstcl |
⊢ ( ( 𝑆 ∈ CHStates ∧ 𝐵 ∈ Cℋ ) → ( 𝑆 ‘ 𝐵 ) ∈ ℋ ) |
7 |
6
|
ad2ant2r |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( 𝑆 ‘ 𝐵 ) ∈ ℋ ) |
8 |
|
hstorth |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ) |
9 |
|
normpyth |
⊢ ( ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑆 ‘ 𝐵 ) ∈ ℋ ) → ( ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 → ( ( normℎ ‘ ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) ) ) |
10 |
9
|
3impia |
⊢ ( ( ( 𝑆 ‘ 𝐴 ) ∈ ℋ ∧ ( 𝑆 ‘ 𝐵 ) ∈ ℋ ∧ ( ( 𝑆 ‘ 𝐴 ) ·ih ( 𝑆 ‘ 𝐵 ) ) = 0 ) → ( ( normℎ ‘ ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
11 |
5 7 8 10
|
syl3anc |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( normℎ ‘ ( ( 𝑆 ‘ 𝐴 ) +ℎ ( 𝑆 ‘ 𝐵 ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) ) |
12 |
3 11
|
eqtrd |
⊢ ( ( ( 𝑆 ∈ CHStates ∧ 𝐴 ∈ Cℋ ) ∧ ( 𝐵 ∈ Cℋ ∧ 𝐴 ⊆ ( ⊥ ‘ 𝐵 ) ) ) → ( ( normℎ ‘ ( 𝑆 ‘ ( 𝐴 ∨ℋ 𝐵 ) ) ) ↑ 2 ) = ( ( ( normℎ ‘ ( 𝑆 ‘ 𝐴 ) ) ↑ 2 ) + ( ( normℎ ‘ ( 𝑆 ‘ 𝐵 ) ) ↑ 2 ) ) ) |